Sidelobe elimination for generalizedsynthetic discriminant functions by atwo-filter correlation and subsequentpostprocessing of the intensity distributions
Mario Montes-Usategui, Juan Campos, Ignasi Juvells, and Santiago Vallmitjana
One of the most important problems in optical pattern recognition by correlation is the appearance ofsidelobes in the correlation plane, which causes false alarms. We present a method that eliminatesidelobes of up to a given height if certain conditions are satisfied. The method can be applied to anygeneralized synthetic discriminant function filter and is capable of rejecting lateral peaks that are evenhigher than the central correlation. Satisfactory results were obtained in both computer simulationsand optical implementation.
1. Introduction
The design of filters that use an optical correlator forpattern recognition has undergone great develop-ment during the past few years. The matched fil-ter,' introduced by VanderLugt in 1964, consists ofthe Fourier transform of the object to be recognized.This filter gives the maximum signal-to-noise ratio inthe correlation plane but is unable to discriminatebetween similar objects. The discrimination capabili-ties and the light efficiency can be enhanced with aphase-only filter,2 whose main drawback is its sensitiv-ity to noise.
The response of these filters depends on the scale,the orientation, and in general any deformation in theinput pattern. A possible solution is to expand thereference objects into a set of orthogonal functionsthat are invariant to one of these deformations. Forinstance, rotation invariance can be achieved with acircular-harmonic expansions (CHE). The informa-tion about the object to be recognized and the objectto be discriminated against can be introduced simulta-neously by means of synthetic discriminant filters4
(SDF's).
The authors are with the Laboratori d'Optica, Departament deFisica Aplicada i Electronica, Universitat de Barcelona, Diagonal647, Barcelona 08028, Spain.
Received 1 June 1993; revised manuscript received 4 November1993.
The main shortcoming in the last two designs is theappearance of sidelobes caused by the loss of informa-tion in the circular-harmonic-expansion case and thelack of control over the whole correlation plane in theSDF approach. Several partial solutions to this prob-lem can be found in the literature. 5 6
The SDF filters are designed so that the centralcorrelation with the training set takes prespecifiedvalues. This can be though of as a set of K equationswith N unknowns, where K is the number of objectsin the training set and N is the number of compo-nents in the filter. Such a system has infinite solu-tions, of which the SDF is a particular one, formed bya linear combination of the training images. TheN-K degrees of freedom in the system may be used tooptimize a performance criterion giving a family offilters known as generalized SDF's.7 The minimumaverage correlation energy (MACE) filter8 is a particu-lar solution that minimizes the energy in the correla-tion plane, thus providing sharp peaks and reducingthe sidelobes. There are other filters such as en-tropy optimized filters9 or maximum-discriminationfilters' 0 that are also designed to control the wholecorrelation plane and to produce sharp correlations.
In this study we present a method to eliminatesidelobes, provided certain conditions are met. Theprocess consists of the use of two filters and thesubsequent postprocessing of the correlation distribu-tions. The two filters are obtained by a new filterbeing established that modifies the whole correlationplane except the central point. This new filter is
then either added to or subtracted from the filterbeing corrected.
The remainder of the paper is organized as follows.In Section 2 we introduce the method and develop themathematical expression of the filter. In Section 3the validity conditions are discussed and a perfor-mance analysis is carried out. The results of acomputer simulation of the method are presented inSection 4, together with several examples. In Sec-tion 5 we present the results of the optical implemen-tation of the method, and in Section 6 some remarksand conclusions are made.
2. Method
A. Theoretical Considerations
If we perform the correlation of an input image withtwo differently designed filters restricted to producethe same central correlation, we can postprocess theoutput distributions in order to improve the results.For example, if we binarize them by applying a giventhreshold and then multiply the output planes pixelby pixel (which is equivalent to processing them withthe logical operation AND), we can eliminate thesidelobes that are more than the threshold value thatare not common to both planes, and we can maintainthe central peak.
The existence of false alarms in such a proceduredepends on the appearance of common sidelobes.The method we present ensures, in certain condi-tions, that no sidelobe is common to both correlationplanes.
The idea of an image being processed by means oftwo digital or optical processes and of the final imagebeing obtained by the pointwise multiplication ofeach single output has been applied in the past invarious problems. The procedure is similar to opera-tions commonly used in mathematical morphology.Casasent et al.," using the correlation with twodifferent filters, detected simple geometrical shapesimmersed in high clutter and noise. The procedureinvolved the binarization of the correlations and thepixel-by-pixel multiplication of both results. Themethod was an optical implementation of the morpho-logical hit-miss transform. More recently, Crowe etal.' 2 proposed the utilization of a similar method toreduce the sidelobes appearing in imaging systemsowing to the finite size of the pupil, thus improvingthe spatial resolution.
We restrict our study to the case in which we obtainreal-valued correlations. The general case of havingcomplex distributions (such as those produced bycircular-harmonic-expansion filters) is treated brieflyin Section 6.
The method can be applied to a wide variety offilters (which is referred to as the base filters in whatfollows) and consists of addition and subtraction of anew filter (called the correcting filter), designed sothat the following conditions are fulfilled:
(I) The filter is orthogonal to every image in thetraining set (the filter and the images are treated as
vectors with the usual lexicographic ordering). Thisrequirement ensures that the central correlationsproduced by the base filter remain unchanged.
(II) The correlation between the correcting filterand the images produces a constant plane with apredefined value. This condition is accomplishedonly in an approximate form by means of a Lagrangeminimization process.
The output distributions obtained with the newfilters have two terms:
(hb + he) *xi = hb *Xi + hc*Xi . (1)
(hb - hc) * xi hb * Xi - he * Xi' (2)
where hb is the base filter, he is the correcting filter, xiis one of the images in the training set, and thesymbol * means correlation.
The expression hb * xi is real and may take positiveand negative values. The term h * x in Eq. (1),which is constant over the whole plane, increases thepositive sidelobes and decreases the negative ones.Conversely in Eq. (2), he * xi increases the negativesidelobes and decreases the positive ones. A suitablechoice of the value of the constant plane and thethreshold ensures that no sidelobe is common to bothbinarized correlations. These considerations are dis-cussed in detail in Section 3.
Although we put the emphasis on generalized SDFfilters because of the practical importance of thesedesigns, the method can be applied to other filterswith minor modifications. The only condition thatis necessary for the base filter is to show a gooddiscrimination between the true and the false classessince the only point that is not changed is the centralcorrelation. Therefore the method can be used inconjunction with filters that are already designed toavoid sidelobes, such as the entropy optimized filters9
or the maximum-discrimination filters,10 to achievehigher discrimination capabilities.
B. Filter Design
Let x,(j), x2(j), .. ., Xk(j) be the k training images ofN components (j= 1,... , N), and let X,(w),...,Xk(w) denote their Fourier transforms. Let H,(w) bethe Fourier transform of the correcting filter.
Condition (I) in Subsection 2.A can be written asfollows:
N
z He(w)X8(w) = .w=l
i= 1,...,k.
Condition (II), as mentioned above, can be achievedonly approximately by minimization of the followingerror function:
1 K NE = K I D(w) - He(w)X(w)I 2, (4)
Kit, w=1
where D(w) represents the Fourier transform of thedesired shape for the correlation between the images
and the filter (a plane in our case). Expression (4) istherefore a measure of the mean error between thecorrelations obtained and those desired. This filteris a particular case of the minimum-squared-errorsynthetic discriminant function'3 design introducedby Vijaya Kumar et al.
Expression (3) can be written compactly as
by means of a Lagrange optimization process. Inexpression (13), X denotes a K-dimensional complexvector containing the Lagrange multipliers.
By calculating the gradients with respect to thefilter components and the Lagrange multipliers andby setting them to zero, we obtain the followingexpression for the correcting filter (the mathematicaldetails can be found elsewhere'3 ):
S+h, = 0, (5)
where h is the N-dimensional vector whose compo-nents are H,(w)
h = [Hc(1), HC(2), .. ., H(N)]T. (6)
S is an N x K matrix formed by the Fourier trans-forms of the images arranged in columns:
X1(2)SX=
X(N)
X2 (1)
X2 (2)
X2 (N)
XK(1)K1... XK(2)
... XK(N)J
The superscript + means the conjugate transpose ofthe matrix. Finally, 0 represents the K-dimensionalvector with all its components zero.
We can express Eq. (4) with the same formalism bydefining the N x N diagonal matrix,
-Xi(1) 0
0 Xi (2)Pi = 0
0 0
..~~ ~~~~. .and the N-dimensional vector,
d = [D(1), D(2), . . ., D(N)]T.
With such definitions the error can be written as
(9)
1 X
E = K [(d - Phc)+(d - Phj)]
= d+d - h+r - r+h, + h+Phc, (10)
where
r = K 2 (Pid), (11)i=l
K (P APE(12)
Conditions (I) and (II) can be accomplished simulta-neously by minimization of
L[h,] = (d+d - h r - r+hc + hPh,) - 2X+(S+Hc)
h = [I - P-'S(S+P-'S)'1S+]P-r. (14)
3. Necessary Conditions and Performance Analysis
Let us suppose we have designed a generalized SDFthat solves a two-class problem. The prespecifiedvalues for the correlation with classes A and B arecalled po and pl, respectively, and with no loss ofgenerality we suppose that po > pl. The criterionfor classification of an image as a member of one ofthe two classes is the following: If the correlationintensity at the center exceeds a given threshold, theimage is classified as belonging to class A; otherwise,the image is assigned to class B.
As commented on in Section 2, the method forelimination of sidelobes involves the correlations withtwo filters that have an opposite effect. The first,which we call the positive filter, reduces the negativesidelobes and enhances the positive ones. The sec-ond filter, called negative, reduces the positive peaksand enhances the negative ones. Our goal is todetermine the proper settings for the threshold andthe constant plane resulting from the correlationsbetween the images and the correcting filter in such away that the only point in the output intensitydistributions that passes the threshold in both casesis the central peak.
The equations that ensure the above statement canbe written as follows:
O( lX + C)2 < po
(lXi - )2 < 0p2
C2 < 0po
p2 < op2p1
(15)
(16)
(17)
(18)
where 0 is a factor between zero and one thatrepresents the threshold, c is the value of the con-stant plane, and x is the height of the maximumsidelobe to be suppressed.
The necessity of conditions (15)-(18) is discussed inthe following considerations. Maximum sidelobe xincreased by constant c may become higher thanvaluepo of the correlation for class A. This situationis illustrated in Fig. 1. In Fig. 1(a) the positivesidelobe (x > 0) is increased by positive constant c.The resulting intensity can be seen in Fig. 1(b), inwhich it appears higher than the intensity of thecentral correlation p'. For negative sidelobes wehave an equivalent situation. In Fig. 1(c) the nega-tive sidelobe (x < 0) is increased (in absolute value) bynegative constant -c, and the resulting intensity
Fig. 1. (a) Amplitude of the correlation plane produced by thebase and the correcting filters when both are added (positivefilter). The dashed curves represent the constant amplitudecorrelation value given by the correcting filter. The solid linesrepresent the correlation peaks (po andpi) and the sidelobes (x > 0andx <0) obtained with the base filter. (b) Intensity distributionfor the positive filter. (c), (d) Same as (a) and (b) for the case of thenegative filter.
matter whether the maximum sidelobe is positive ornegative. In other words, this factor takes intoaccount the case (x,- )2 for negative peaks and(x + c)2 for positive ones.
Inequality (16) represents the condition for thedecreased maximum sidelobe to be eliminated in thebinarization; i.e., when the absolute value of thesidelobe is decreased, the resulting intensity has to belower than the limit marked by the threshold value[see Figs. 1(a) and 1(b) forx < 0 and Figs. 1(c) and 1(d)for x > 0]. The condition is written assuming thatthe threshold is applied on po, but the maximumvalue in the output intensity distribution might bedifferent. For instance, if we had two maximumsidelobes, both with the same absolute value but withdifferent sign, and if the increased term (lxi + c)2
were higher than po, the threshold value would be0(lx + )2 because when one decreases, the otherincreases (the case represented in Fig. 1). By setitnga more restrictive condition, such as Eq. (16), we cancover all the cases.
In addition, we need to ensure that the sidelobeslower than x disappear. We have to treat two casesseparately:
(a) If c < x 1/2, then (Ix - )2 > (x I - Ix /2) 2 >c2. In this situation, inequality (16) guarantees theelimination of every sidelobe in the range from zero tox [see Figs. 2(a) and 2(b)].
(b) If c > Ix 1/2, the former condition is not assured,and small sidelobes may surpass the threshold. Thissituation is depicted in Figs. 2(c) and 2(d).
Inequality (17) is then necessary to take into ac-count case (b). Finally, inequality (18) is requiredfor a correct classification of class B.
In the typical case in which p0 = 1 and pi = 0,inequalities (15)-(18) become
lXI - 01/2 < < 01/2 -lX, (19)
C < 01/2 (20)
As can be observed in the above expressions, thevalue of c is not completely determined by the condi-tions, but there is a range of permissible values that isnecessary for the reliability of the method. Thisnecessity is caused by, as mentioned above, thecorrelation planes with the correcting filter not beingexact planes but approximate versions obtained bymeans of a minimization process.
The maximum sidelobe that fulfills expression (19)is obtained when
lxi - 01/2 = 0-1/2 - IXI; (21)
[Fig. 1(d)] is also higher than the central correlationfor class A. Inequality (15) is then necessary toguarantee that po passes the threshold. The expres-sion ( x + c)2 is the intensity of the increased peak no
whence
lxI = ( + 1)0-1/2/2. (22)
On the other hand, by considering inequality (20) and
------- 4- - --------@________~~___________________________(x-C)2 T 2sc
(b)
o
C '-------------1 r-------------_
x/2.
____ ~ ~ It ST
|J L
-c ------------- L_______________(c)
2
(s-c)~
(- ).._______________ _____________
(x-c) _~~~P
(d)
Fig. 2. Illustration of the necessity of inequality (17): (a) Casec < x/2. The solid lines represent the amplitude of the correla-tion with the base filter, which produces a high (peakx) and a small(peak s) sidelobe. (b) Intensity distribution corresponding to(a). If c < x/2, sidelobes lower than x are suppressed by Eq.(16). (c) The same as (a) when c > x/2. (d) Intensity distributioncorresponding to (c). When c > x/2, small sidelobes increased bythe constant may surpass the threshold value.
the leftmost part of inequality (19), or
C < 1/2,
C > Xi - 01/2,
we obtain the maximum Ix when
lxi - 01/2 = ol/2, (23)
(24)
Therefore the maximum sidelobe can be written as
IxImn,. = min[201/2 , (0 + 1)0-1/2/2]. (25)
The optimum threshold, 0, would be
201/2 = (0 + 1)0-1/2 = 0 = 1/3,
and finally x Imax = 1.15 (115% of the central correla-tion) in amplitude or Ix12i = 1.32 (132% of thecentral correlation) in intensity.
In practice it is not possible to reach this limitbecause the range of permissible values for c ininequality (19) is reduced to a single point. How-ever, in most practical situations the method enablesthe elimination of sidelobes higher than the centralpeak, which cannot be corrected by binarization ofthe output intensity produced by the base filter with asingle threshold.
Figure 3 represents the permissible variation ofconstant c (shaded area) when the threshold value isfixed (p0 = 1,p, = 0, and 0 = 0.5). The graph showsthat for small variations in c, large sidelobes can beeliminated, but for wide variations the height of thesidelobe must be small. By varying the thresholdand by fixing the desired height of the sidelobe to besuppressed, we obtain Fig. 4. ( x = 1). As can beobserved, there is a value of 0 for which the permis-sible range of c is maximum because of the monotonicbehavior of the restrictions in inequalities (19) and(20). This optimum threshold can be calculated byuse of the following equation:
01/2 - 0-1/2 - I X Iop op (26)
Consequently we can determine all the parameters
1
c=(1/0.5)- xI
0.80-
a)
C0 0.60a)
.Z _
.2 0.40Ea)
0.20
0.00 . 60.0 0.1 0.2 0.3 0.4 0.5Maximum sidelobe
0.6 0.7 0.8 0.9 1height (=0.5)
Fig. 3. Permissible range for constant c as a function of themaximum sidelobe to be eliminated (po = 1,pl = 0, 0 = 0.5).
Fig. 4. Permissible range for constant c as a function of thethreshold (po = 1, pi = 0, and l x = 1).
envisaged by the method by selecting the maximumheight of the sidelobe to suppress, by calculating theoptimum threshold by means of Eq. (26), and finallyby choosing constant c as the midpoint value of therange given by inequality (19).
The validity of the method is determined by theextent to which the actual correlations between theimages and the correcting filter are constant planeswith the expected accuracy. The possibility of thesidelobes being eliminated in the range from 0 to aprespecified value, x , depends on whether the mini-mization procedure is capable of producing a correct-ing filter so that the correlations obtained with theimages in the training set satisfy inequality (19).Therefore the performance of the method is deter-mined by the deviations in the correlation distribu-tion from the expected plane; in other words, themaximum sidelobe that can be suppressed depends onthe range of variation of the points in the correlationplane. On the other hand, the height of the side-lobes depends on the similarity between images indifferent classes; i.e., the more similar the imageswith different conditions are, the larger the, expectedvalues of the sidelobes become.
The method gives more accurate correlations planes,i.e., the correcting power is higher, when the imagesin the training set are more similar and thereforewhen higher sidelobes appear. To demonstrate thisproperty, let us assume we have two N-dimensionalimages whose Fourier transforms areX 1(w) andX2 (w).
The expression for the error in Eq. (4) can then bewritten as
1 rNE= I I D(w) - HC(W)Xl(W) 1
2
2 Lw=1
N+ T ID(w) - Hc(w)X* (w) 12 (27)
w=l
E = 1/2[(d - P hj)+(d - P *hj)
+ (d - P2h,)+(d - P h,)]. (28)
Let us suppose that P, and P2, which are diagonalmatrices, can be inverted (i.e., there is no frequencyfor which the Fourier transform of the images iszero), and let us define filters h, and h2 so that
(29a)
h2 = (P*2)-ld; (29b)
h, and h2 represent the filters that give exactly thedesired shape when they are correlated with imagesX, and X2, respectively. The assumption that P1 andP2 are invertible is not so restrictive, and similarrequirements are needed in other filter designs. Inparticular, in MACE filters, the matrix that repre-sents the average energy of the images in the trainingset must also be full rank.
Because h, is the optimum filter, by substituting h,for h, in Eq. (28) we have
E < 1/2[(d - Phl)+(d - P hl)
+ (d - P hl)+(d - P hl)]
= 1/2[(d - P'2hl)+(d - P hl)]
= 1/2[(P h2 - P hl)+(P h2 - P hl)]
= /2 {[P*(h2 - hl)]+P (h2 - )}, (30)
and from Eqs. 29(a) and 29(b),
P*h 2 - P*hl = 0. (31)
If we express image X2 as a function of Xl, we canwrite
P2 = P1 + A, (32)
and therefore
(P* + A*)h2 - P*hl = 0, (33)
P*(h2 - hj) = - A*h2
= -. A*(P* + A*)-ld. (34)
As P, is a full-rank matrix, if A tends to zero, then(h2 - hj) tends to zero, and in consequence the errorin inequality (30), which depends on this difference,becomes increasingly small:
A >0, P2 Plart(h2 - h) >0 --E ->. Q.E.D.
We carried out an experimental verification of thisproperty. The details are given in Section 4.
Fig. 5. Sequence of images used in the simulation. Letter a isab(0), letter b is ab(10), and the intermediate patterns are ab(i),withi = 1, . . ., 9.
4. Computer ExperimentsIn this section we present the results of severalexperiments carried out by means of a computersimulation in order to test the suitability of themethod in practical situations. We performed astudy of the dependence between the value of theexpected sidelobes and the correcting capabilities ofour method by using the images depicted in Fig. 5.A set of ten correcting filters was designed, each ofthem calculated by use of a pair of images from thesequence ab(O)-ab(i); namely, filter 1 was calculatedwith ab(O) and ab(1), filter 2 with ab(O) and ab(2), andso on. The measure of the similarity between imageab(i) and ab(O) was calculated with the followingexpression:
where the symbol * means correlation.The error function in Eq. (4) as well as the devia-
tion from a perfect plane (c = 0.35) for each singleimage was computed for every correcting filter, and
12.00 -L.
0a)
0 8.00-a)L.00-(F)
4.00 -
0.40 0.50 0.60 0.70 0.80 0.90 1.00
Si(ea b(), b (i))
Fig. 6. Squared error as a function of similarity between images.
Fig. 7. al, a2, Images used to design the MACE filter. Theimposed values were 1 for image al and 0 for image a2. bl, b2,Intensity of the correlation betweeen the MACE filter and imagesal and a2, respectively. cl, c2, Intensity of the correlationbetween the positive filter and images al and a2. dl, d2, Intensityof the correlation between the negative filter and images in al anda2. el, e2, Same as images cl and c2 binarized with 0 = 0.36. fl,f2, Same as images dl and d2 binarized with 0 = 0.36. gl, Resultof pixel-by-pixel multiplication of images el and fl. g2, Result ofpixel-by-pixel multiplication of images e2 and f2.
the results are represented in Fig. 6. The graphshows the dependence between these deviations fromthe expected shape and the similarity measure givenby Eq. (35). For very similar images such as ab(O)
Fig. 8. Images used to design the filters. The imposed values forthe central correlations were 1 for S, 1 for C, and 0 for E.
*444'* Error for O)~ -r ro fn ;N fob(O) v_ cbru . u ..,b(O) ***r*Average error
Av. ,A
abtl) .N1
w\&~
1t f; nn
y
and ab(1) S[ab(0), ab(1)] = 0.96} the error is small(E = 0.25), and conversely, when the similarity be-tween images is small {S[ab(0), ab(1)] = 0.57 for ab(0)and ab(10)}, the error is high (E = 10.3). The resultshows the expected behavior; i.e., when large side-lobes are more likely to appear, owing to the similar-ity between images with different constraints, theprocedure we propose is more powerful because of asmaller variation with respect to the desired plane.
The increasing correcting power enables the elimi-nation of sidelobes, even if they are higher than thecentral peak, as illustrated in Fig. 7. In Fig. 7, twoimages with a similarity S = 0.90 were used to build a
MACE filter by imposition of images (al) and (a2) togive values of 1 and 0, respectively. The MACE filteris an antisidelobe design, but in this situation it givesseveral lateral peaks, the largest of which has a valueof 126% of the central correlation (in intensity), asshown in images (bi) and (b2).
In order to eliminate the sidelobe, we prepared thecorrecting filter using the following parameters:
which are specifically designed to avoid the appear-ance of sidelobes. However, there is no noise resis-tance included in the minimum-squared-error SDFfilter design, so the procedure is highly sensitive tonoisy inputs.
The possible solutions to this problem are the sameas those used to introduce noise resistance in theMACE design: trade-off filters'4 and the modifica-tion used in minimum noise and correlation energyfilters.' 5 A further study on the suitability of thesesolutions will be carried out in the future.
5. Optical Results
The method proposed for elimination of sidelobes wastested by use of a convergent correlator.' 6 Thissetup has an advantage in that it permits easymatching between the scales of both the input imageand the filter.
The filters were built by means of computer-generated holograms codified by Burkhardt'smethod,' 7 displayed on a laser printer, and photore-duced. The holograms were sandwiched to avoiduncontrolled phases owing to thickness variations inthe photographic film. A low-power He-Ne laserprovided the coherent illumination. Finally, a CCD
Fig. 9. (a), (b), (c) Intensity of the correlation between thecomposite filter and letters S C, and E, respectively. (d), (e), (f)Same as (a), (b), and (c) with the positive filter. (g), (h), (i) Same as(a), (b), and (c) with the negative filter.
The results of the correlations between the positiveand the negative filters are shown in images (cl)-(d2)of Fig. 7, and the binarized results are shown inimages (el)-(f2). After pixel-by-pixel multiplicationof both binarized planes, images (gl) and (g2) wereobtained. As can be observed, all the sidelobes aresuppressed, and a perfect detection of the centralcorrelations is possible.
The method is then capable of producing a signifi-cant increase in the discriminant abilities of the SDFfilters, including those such as the MACE design,
(a) (d) (g)
(b) (e) (h)
(c) (f) (i)
Fig. 10. (a), (b), (c) Intensity of the correlation between thepositive filter and letters S, C, and E, respectively, binarized with0 = 0.4. (d), (e), (f) Same as (a), (b), and (c) with the negativefilter. (g) Result of pixel-by-pixel multiplication of images (a) and(d). (h) Result of pixel-by-pixel multiplication of images (b) and(e). (i) Result of pixel-by-pixel multiplication of images (c) and (f).
camera and a frame grabber were used to capture theresulting correlation distributions.
The images that were used in the design of thefilters are shown in Fig. 8. The values imposed forthe correlation at the origin were 1, 1 0, for S, C, andE, respectively. The correlations between the threeletters and a composite filter are shown in Figs.9(a)-9(c), in which large sidelobes can be observed.
The correlations obtained with the positive and thenegative filters are depicted in Figs. 9(d)-9(i). As canbe seen, the effects of the two filters are opposite; thesidelobes reduced by the first one are enhanced by theother and vice versa, as expected. By binarizing theresults obtained with the positive and the negativefilters with a threshold of 0.40, we obtain the imagesin Figs. 10(a)-10(f). Finally, in Figs. 10(g)-10(i) theresults of pixel-by-pixel multiplication of binarizedimages are represented.
The results observed in the optical implementationwere satisfactory and showed good agreement withprevious computer simulations. Therefore themethod seems to be suitable for application in apractical situation.
6. Final Remarks and Conclusions
The existence of lateral peaks is one of the mostimportant problems in optical pattern recognition bymeans of correlation. In this study we present amethod that eliminates every sidelobe within a givenrange, provided certain conditions are fulfilled. Themethod has the following properties:
* It can be applied to a wide variety of filters.* The method ensures the elimination of the
sidelobes if certain conditions are satisfied.* Sidelobes higher than the central correlation
can be suppressed.* The method is more powerful when higher
sidelobes are expected.
The procedure was tested with simulation andoptical implementation and gave satisfactory resultsin both cases. However, some limitations were found.In Section 4 the noise sensitivity of the filter waspointed out. A possible way to reduce this drawbackis by means of a compromise filter.14 The underlyingidea of this type of design is to introduce a new termin the error function to be minimized, which repre-sents the output variance of the noise. A parameteris used to balance the importance of the two conflict-ing goals. This solution represents a trade-off be-tween the noise resistance and the minimization withrespect to the desired shape, which thus affects theheight of the sidelobes that can be eliminated by theprocedure. A similar idea, which also involves acompromise between the two magnitudes to be mini-mized, is applied in minimum noise and correlationenergy filters.15 A further study of the optimumselection for this trade-off is needed.
On the other hand, the method we present is valid
only for filters that produce correlations distributionswith real values. The procedure may be generalizedto the case in which complex distributions are ob-tained, by use of a battery of filters, each producing aconstant complex-valued plane whose phases aredistributed over the entire unit circle.
The greater the number of filters used, the greaterthe height of the lateral peaks that can be eliminated,because the directions of the opposing vectors ap-proach 180° with respect to the direction of thesidelobe. However, an increasing number of filtersimplies a more complex procedure. The study of theminimum number of filters required for eliminationof sidelobes with a given height is currently inprogress.
This work has been supported in part by theSpanish Comisi6n Interministerial de Ciencia y Tec-nologia under project ROB91-0554.
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